Abstract
In the brane world scenario, possessing the self-tuning property of the cosmological constant, we study the probability amplitude for an initial state of the Universe to transform to final states with some curvatures. In the Kim, Kyae, and Lee model, there exists a finite range of parameters such that the transition amplitude to a near flat universe is exponentially dominated by the smallness of curvature, $\ensuremath{\gtrsim}{e}^{\mathrm{positive}\text{ }\mathrm{number}/{\overline{\ensuremath{\Lambda}}}^{2}}$, i.e., it is dominated by an almost flat universe of $|\overline{\ensuremath{\Lambda}}|<ϵ$.
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